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In set theory and other branches of mathematics, (pronounced beth two), or 2c (pronounced two to the power of c), is a certain cardinal number. It is the 2nd beth number, and is the result of cardinal exponentiation when 2 is raised to the power of c, the cardinality of the continuum.
- The power set of the set of real numbers, so it is the number of subsets of the real line, or the number of sets of real numbers;
- The power set of the power set of the set of natural numbers, so it is the number of sets of sets of natural numbers;
- The set of all functions from the real line to itself;
- The power set of the set of all functions from the set of natural numbers to itself, so it is the number of sets of sequences of natural numbers;
- The set of all real-valued functions of n real variables to the real numbers.
Some early set theorists hypothesised the equation
stating that 2c is equal to the 2nd aleph number. It turns out that the truth of this equation (*) cannot be determined from the standard Zermelo-Fraenkel axioms of set theory; it is true in some models and false in others. (*) is a part of the generalized continuum hypothesis (GCH), but it is possible that (*) is true while the full GCH is false. On the other hand, if (*) is true, then the ordinary continuum hypothesis (CH) must follow, but again it is possible that CH is true while (*) is false.